# How do you use the Intermediate Value Theorem and synthetic division to determine whether or not the following polynomial #P(x) = x^4 + 2x^3 + 2x^2 - 5x + 3# have a real zero between the numbers 0 and 1?

##### 1 Answer

You can't.

#### Explanation:

You can use synthetic division (and the Remainder Theorem) to find

and

If we one of these positive and the other negative, then, since

The Intermediate Value Theorem can never be used to conclude that there is not a zero in an interval.

We could conclude that

To help explain why, here is the graph of a **different** polynomial function:

graph{ x^4 + 2x^3 + 2x^2 - 5x +1 [-3.067, 3.09, -1.17, 1.91]}

Call this polynomial function

Notice that **does** have a zero in